\(\int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx\) [8]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=-\frac {a^4 c^3}{3 x^3}+\frac {a^3 b c^3}{x^2}-b^4 c^3 x+2 a b^3 c^3 \log (x) \]

[Out]

-1/3*a^4*c^3/x^3+a^3*b*c^3/x^2-b^4*c^3*x+2*a*b^3*c^3*ln(x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=-\frac {a^4 c^3}{3 x^3}+\frac {a^3 b c^3}{x^2}+2 a b^3 c^3 \log (x)-b^4 c^3 x \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^4,x]

[Out]

-1/3*(a^4*c^3)/x^3 + (a^3*b*c^3)/x^2 - b^4*c^3*x + 2*a*b^3*c^3*Log[x]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (-b^4 c^3+\frac {a^4 c^3}{x^4}-\frac {2 a^3 b c^3}{x^3}+\frac {2 a b^3 c^3}{x}\right ) \, dx \\ & = -\frac {a^4 c^3}{3 x^3}+\frac {a^3 b c^3}{x^2}-b^4 c^3 x+2 a b^3 c^3 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=c^3 \left (-\frac {a^4}{3 x^3}+\frac {a^3 b}{x^2}-b^4 x+2 a b^3 \log (x)\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^4,x]

[Out]

c^3*(-1/3*a^4/x^3 + (a^3*b)/x^2 - b^4*x + 2*a*b^3*Log[x])

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80

method	result	size
default	\(c^{3} \left (-b^{4} x +2 a \,b^{3} \ln \left (x \right )-\frac {a^{4}}{3 x^{3}}+\frac {a^{3} b}{x^{2}}\right )\)	\(36\)
risch	\(-b^{4} c^{3} x +\frac {a^{3} b \,c^{3} x -\frac {1}{3} a^{4} c^{3}}{x^{3}}+2 a \,b^{3} c^{3} \ln \left (x \right )\)	\(44\)
norman	\(\frac {a^{3} b \,c^{3} x -\frac {1}{3} a^{4} c^{3}-b^{4} c^{3} x^{4}}{x^{3}}+2 a \,b^{3} c^{3} \ln \left (x \right )\)	\(46\)
parallelrisch	\(\frac {6 a \,b^{3} c^{3} \ln \left (x \right ) x^{3}-3 b^{4} c^{3} x^{4}+3 a^{3} b \,c^{3} x -a^{4} c^{3}}{3 x^{3}}\)	\(50\)

[In]

int((b*x+a)*(-b*c*x+a*c)^3/x^4,x,method=_RETURNVERBOSE)

[Out]

c^3*(-b^4*x+2*a*b^3*ln(x)-1/3*a^4/x^3+a^3*b/x^2)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=-\frac {3 \, b^{4} c^{3} x^{4} - 6 \, a b^{3} c^{3} x^{3} \log \left (x\right ) - 3 \, a^{3} b c^{3} x + a^{4} c^{3}}{3 \, x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^4,x, algorithm="fricas")

[Out]

-1/3*(3*b^4*c^3*x^4 - 6*a*b^3*c^3*x^3*log(x) - 3*a^3*b*c^3*x + a^4*c^3)/x^3

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=2 a b^{3} c^{3} \log {\left (x \right )} - b^{4} c^{3} x - \frac {a^{4} c^{3} - 3 a^{3} b c^{3} x}{3 x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**4,x)

[Out]

2*a*b**3*c**3*log(x) - b**4*c**3*x - (a**4*c**3 - 3*a**3*b*c**3*x)/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=-b^{4} c^{3} x + 2 \, a b^{3} c^{3} \log \left (x\right ) + \frac {3 \, a^{3} b c^{3} x - a^{4} c^{3}}{3 \, x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^4,x, algorithm="maxima")

[Out]

-b^4*c^3*x + 2*a*b^3*c^3*log(x) + 1/3*(3*a^3*b*c^3*x - a^4*c^3)/x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=-b^{4} c^{3} x + 2 \, a b^{3} c^{3} \log \left ({\left | x \right |}\right ) + \frac {3 \, a^{3} b c^{3} x - a^{4} c^{3}}{3 \, x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^4,x, algorithm="giac")

[Out]

-b^4*c^3*x + 2*a*b^3*c^3*log(abs(x)) + 1/3*(3*a^3*b*c^3*x - a^4*c^3)/x^3

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx=-\frac {c^3\,\left (a^4+3\,b^4\,x^4-3\,a^3\,b\,x-6\,a\,b^3\,x^3\,\ln \left (x\right )\right )}{3\,x^3} \]

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x^4,x)

[Out]

-(c^3*(a^4 + 3*b^4*x^4 - 3*a^3*b*x - 6*a*b^3*x^3*log(x)))/(3*x^3)